Fast Numerical Methods for Stochastic Computations: A Review

By Dongbin Xiu. Communications in Computational Physics, 2009

765 citations recorded by the source corpus.

This paper presents a review of the current state-of-the-art of numerical methods for stochastic computations. The focus is on efficient high-order methods suitable for practical applications, with a particular emphasis on those based on generalized polynomial chaos (gPC) methodology. The framework of gPC is reviewed, along with its Galerkin and collocation approaches for solving stochastic equations. Properties of these methods are summarized by using results from literature. This paper also attempts to present the gPC based methods in a unified framework based on an extension of the classical spectral methods into multi-dimensional random spaces. AMS subject classifications: 41A10, 60H35, 65C30, 65C50

Subfields: Civil and Structural Engineering, Statistical and Nonlinear Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/39acf02f2936619d5cbfa05cfce8a4beef0e46c8

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Civil and Structural Engineering

In Statistical and Nonlinear Physics

In Statistics and Probability

Citations in the corpus

Adaptive sparse polynomial chaos expansion based on least angle regression (2011, background): “In the authors’ opinion, quadrature should be preferred to stochastic collocation in practice since: – the computational manipulation of multivariate Lagrange polynomials is cumbersome [57]; – quadrature is based on an explicit representation in a PC basis, which allows straightforward postprocessin”
Proper Generalized Decompositions and Separated Representations for the Numerical Solution of High Dimensional Stochastic Problems (2010, neutral): “[17, 34, 38, 50] and the references therein).”
A multiscale method with patch for the solution of stochastic partial differential equations with localized uncertainties (2013, background): “Numerous developments have been realized for the propagation using functional approaches (see reviews [1, 2, 3, 4]).”
A review of uncertainty analysis in building energy assessment (2018, methodology): “In contrast, non-sampling methods include perturbation methods, moment equations, spectral representations (stochastic Galerkin and collocation) methods, and classical stochastic differential equations [103-105].”
eXtended Stochastic Finite Element Method for the numerical simulation of heterogeneous materials with random material interfaces (2010, background): “In the last two decades, a growing interest has been devoted to spectral stochastic methods [1, 2, 3, 4] for the propagation of uncertainties through physical models governed by stochastic partial di erential equations (SPDEs) involving a nite dimensional noise.”
Nonlinear Propagation of Orbit Uncertainty Using Non-Intrusive Polynomial Chaos (2013, methodology): “This work examines the application of PCEs [13,18,20,27–35] to the solution of Eq.”
A Deep Neural Network Surrogate for High-Dimensional Random Partial Differential Equations (2018, methodology): “. Stochastic calculus methods are being used extensively for classical stochastic PDEs (e.g. [1]). Traditionally, a number of methods are implemented to solve PDEs characterized by random coecients. [2]. Monte Carlo Sampling (MCS) is among one of the most commonly used methods [3]. In MCS, independe”
Convergence Analysis for Stochastic Collocation Methods to Scalar Hyperbolic Equations with a Random Wave Speed (2010, neutral): “, [1, 6, 13, 14]).”
Simulation-based optimal Bayesian experimental design for nonlinear systems (2011, background): “Many options exist, ranging from projection-based model reduction [45, 46] to spectral methods based on polynomial chaos (PC) expansions [47, 22, 23, 48, 49, 50, 51].”
Survey of multifidelity methods in uncertainty propagation, inference, and optimization (2018, background): “We refer to [130, 208, 209, 139, 207] for an introduction to polynomial chaos expansion and further details.”
Surrogate and reduced-order modeling: a comparison of approaches for large-scale statistical inverse problems [Chapter 7] (2010, methodology): “While a survey of polynomial chaos methods for solving ODEs and PDEs with random inputs is beyond the scope of this chapter (see for instance Najm (2009); Xiu (2009)), we highlight two broad classes of approaches.”
Bayesian Solution Uncertainty Quantification for Differential Equations (2013, methodology): “ng use of sampling based methods, perturbation of initial system states (Mosbach and Turner, 2009), moment closure approaches, and more recently polynomial chaos methods (see Ghanem and Spanos, 2003; Xiu, 2009, for a complete overview). The inverse problem concerns the uncertainty in the model input”
Sparse tensor discretizations of high-dimensional parametric and stochastic PDEs* (2011, methodology): “University of Basel Library, on 30 May 2017 at 13:41:54, subject to the Cambridge Core terms of use, available to the survey by Xiu (2009) for further references on various applications of such ‘generalized polynomial chaos’ (GPC) methods.”
A stochastic collocation approach to Bayesian inference in inverse problems (2009, neutral): “For an extensive discussion of gPC-based algorithms, see [26].”
Towards efficient uncertainty quantification in complex and large-scale biomechanical problems based on a Bayesian multi-fidelity scheme (2014, background): “The major disadvantage of solution schemes based on PCE is the curse of dimensionality, which renders these approaches infeasible for problems with high stochastic dimension as is the case when considering random fields.”